Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $\sqrt{6x}\cdot \sqrt{3x^2}$

Verified step by step guidance

1. Identify the two functions being multiplied: here, the first function is \(\sqrt{6x}\) and the second function is \(\sqrt{3x^2}\).
2. Recall that the product rule for derivatives states: \(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\), but since the problem asks to simplify the product, focus on simplifying the expression \(\sqrt{6x} \cdot \sqrt{3x^2}\) first.
3. Use the property of square roots that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) to combine the two square roots into one: \(\sqrt{6x} \cdot \sqrt{3x^2} = \sqrt{(6x)(3x^2)}\).
4. Multiply inside the square root: \((6x)(3x^2) = 18x^3\), so the expression becomes \(\sqrt{18x^3}\).
5. Simplify \(\sqrt{18x^3}\) by factoring inside the root to extract perfect squares: write \$18x^3\( as \)9 \cdot 2 \cdot x^2 \cdot x\(, then use \)\sqrt{a^2} = a$ to simplify.

Key Concepts

Product Rule for Radicals

The product rule for radicals states that the square root of a product equals the product of the square roots, i.e., √a * √b = √(a*b). This rule allows simplification by combining radicals under a single square root when multiplying.

Properties of Exponents

When multiplying expressions with the same base, add their exponents: x^m * x^n = x^(m+n). This property helps simplify terms like x and x^2 inside or outside radicals.

Simplifying Radicals with Variables

To simplify radicals containing variables, express variables with exponents inside the root and apply exponent rules. For nonnegative variables, √(x^2) simplifies to x, ensuring the expression remains valid.